On collination groups of finite planes Supplement to the notes of Arrigo Bonisoli

It is our purpose to specify some of the theory of collineation groups of finite projective planes as introduced by A. Bonisoli (although we do not discuss the complete notes here). We included a lot of relevant exercises of varying degree of difficulty. 1 Notation, definitions and some exercises Points will always be denoted by small latin letters, lines by capitals (unlike Bonisoli’s notations). By (a, b), with a, b ∈ N, we denote the greatest common divisor of a and b. Suppose π is a projective plane of order n, and suppose (p, L) is a point-line pair. Then a collineation θ of π is a (p, L)-collineation if θ fixes any point on L and every line through p. If (p, L) is a flag, then θ is also called a (p, L)elation; if (p, L) is an anti-flag, then θ is said to be a (p, L)-homology. In both cases, L is called the axis and p the center. A collineation of a finite projective plane has a center if and only if it has an axis. Collineations with centers are often called central collineations or perspectivities.